Spread Frequency Shift Keying (S-FSK) is a modulation and demodulation technique that combines some of the advantages of a classic spread spectrum system (for example, resistance against narrowband interferers) with the advantages of a classic FSK system (low-complexity).
In order to better understand this disclosure, a brief review of the S-FSK modulation is reported. An S-FSK transmitter includes a binary FSK transmitter in which the frequency deviation fd is large enough to generate a spectrum with two separate lobes. For this reason, the concept of dual channel is introduced: channel 0 refers to the signal placed around a frequency f0 and channel 1 refers to the signal placed around a frequency f1, with fd=(f1−f0)/2. The symbols to be transmitted are generated with a rate 1/T, where T is the symbol period, and belongs to the alphabet {−1,+1}. Let dk be the symbol to be transmitted at the time instant kT and let χ+ and χ− be the sets of indices where these symbols assume positive and negative values, respectively (i.e. χ+={k:dk=+1} and χ−={k:dk=−1}). The transmitted signal is
                              s          ⁡                      (            t            )                          =                  {                                                                      A                  ⁢                                                                          ⁢                                      sin                    ⁡                                          (                                              2                        ⁢                        π                        ⁢                                                                                                  ⁢                                                  f                          0                                                ⁢                        t                                            )                                                                                                                                        if                    ⁢                                                                                  ⁢                                          ⌊                                              t                        /                        T                                            ⌋                                                        ∈                                      χ                    -                                                                                                                        A                  ⁢                                                                          ⁢                                      sin                    ⁡                                          (                                              2                        ⁢                        π                        ⁢                                                                                                  ⁢                                                  f                          1                                                ⁢                        t                                            )                                                                                                                                        if                    ⁢                                                                                  ⁢                                          ⌊                                              t                        /                        T                                            ⌋                                                        ∈                                      χ                    +                                                                                                          (        1        )            where A is a real constant and └a┘ is the integer part of a.
A frequency selective channel with an additive non-white Gaussian noise is considered; however, the channel gain Gi and the noise power spectral density Ni are assumed to be flat around the frequency fi. Therefore, at the receiver input, the signal-to-noise ratio (SNR) for the channel i is:
                                          Γ            i                    =                                                    A                2                            ⁢                                                G                  i                  2                                /                2                                                                    N                i                            /              T                                      ,                                  ⁢                              with            ⁢                                                  ⁢            i                    ∈                                    {                              0                ,                1                            }                        .                                              (        2        )            
The SNRs defined in (2) adequately characterizes the quality of the received signal. Moreover, another characterization of the quality of the received S-FSK signal may be made through the unbalancing factor x and the average signal-to-noise ratio Γ. This last term is defined according to “Spread frequency shift keying,” T. Schaub, IEEE Trans. Commun., vol. 42, no. 2, pp. 1056-1064, February 1994 as the ratio of the signal energy and the average noise power densities. These parameters are related to (2) as follows:
                              Γ          =                      2            ·                                                            Γ                  0                                ·                                  Γ                  1                                                                              Γ                  0                                +                                  Γ                  1                                                                    ⁢                                  ⁢        and        ⁢                                  ⁢                  x          =                                                    Γ                1                                            Γ                0                                      .                                              (        3        )            
Let us suppose that correct frame synchronization has been established with any technique and that it is helpful to discriminate at the receiver side whether a +1 or a −1 has been transmitted.
At the receiver side, the signal is processed along two different paths depicted in FIG. 1: on the first path, at the time t a multiplication by the complex exponential exp(−j2πf0t) is performed (corresponding to a frequency shift of f0 Hz in the frequency domain); while, on the second path, a multiplication by the complex exponential exp(−j2πf1t) is performed. Successively, on each path the signal is filtered with a low-pass filter with cut-off frequency I/T/2, where I is the oversampling factor.
Hereinafter, without loosing generality, the low-pass filter LPF is assumed to be ideal. The filter output yi(t), with iε{0,1}, is sampled with a rate I/T and fed into an envelop detector, which is implemented with a weighed sum.
Let {ωn} be the normalized weight coefficients
            (                                    ∑                          n              =              0                                      I              -              1                                ⁢                      ω            n                          =        1            )        ⁢                  ⁢    with    ⁢                  ⁢          E      ω        =            ∑              n        =        0                    I        -        1              ⁢                                                  ω            n                                    2            .      The signal r fed into a receiver may be modeled as follows:
                              r                      0            ,            k                          =                  {                                                                                                            w                                          0                      ,                      k                                                                                                                                                          if                    ⁢                                                                                  ⁢                    k                                    ∈                                      χ                    +                                                                                                                                                                                    w                                              0                        ,                        k                                                              +                                                                  A                        /                                                  2                                                                    ·                                              G                        0                                                                                                                                                                                if                    ⁢                                                                                  ⁢                    k                                    ∈                                      χ                    -                                                                                                          (        4        )                                          r                      1            ,            k                          =                  {                                                                                                            w                                          1                      ,                      k                                                                                                                                                          if                    ⁢                                                                                  ⁢                    k                                    ∈                                      χ                    -                                                                                                                                                                                    w                                              1                        ,                        k                                                              +                                                                  A                        /                                                  2                                                                    ·                                              G                        1                                                                                                                                                                                if                    ⁢                                                                                  ⁢                    k                                    ∈                                      χ                    +                                                                                                          (        5        )            where wi,k is an additive circularly Gaussian noise with zero mean and variance σi2=EωNiI/T, with iε{0,1}.
In the following analysis, the detailed receivers have the same structure. Let ρi denote the value assumed by the signal ri,k. The receiver elaborates ρ0 and ρ1 in order to detect which symbol has been transmitted. This operation is performed in two steps. In the first step, two decision values μ+1(ρ0,ρ1) and μ−1(ρ0,ρ1) are computed. Successively, accordingly to the higher decision value, the detected symbol {circumflex over (d)}k is:
                                          d            ^                    k                =                  {                                                                      +                  1                                                                                                  if                    ⁢                                                                                  ⁢                                                                  μ                                                  +                          1                                                                    ⁡                                              (                                                                              ρ                            0                                                    ,                                                      ρ                            1                                                                          )                                                                              >                                                            μ                                              -                        1                                                              ⁡                                          (                                                                        ρ                          0                                                ,                                                  ρ                          1                                                                    )                                                                                                                                            -                  1                                                            elsewhere                                                                        (        6        )            
S-FSK receivers are based on equation (6), but they differ from each other in the way the decision values are computed. Generally, the decision values μ+1(ρ0,ρ1) and μ−1(ρ0,ρ1) are computed given the knowledge of the channel and the noise parameters, namely the channel gains Gi and the noise variances σi2.
Assuming knowledge of the first LTS symbols at the transmitter and at the receiver sides, which is typical in S-FSK systems, the channel and noise parameters may be estimated using the signals (4) and (5) as follows:
                                          σ            ~                    0          2                =                              1                                                        χ                +                                                            ⁢                                    ∑                              k                ∈                                  χ                  +                                                                                                  ⁢                                                  ⁢                                                            r                  0                                ⁡                                  (                  k                  )                                            2                                                          (        7        )                        and                                                                            σ            ~                    1          2                =                              1                                                        χ                -                                                            ⁢                                    ∑                              k                ∈                                  χ                  -                                                                                                  ⁢                                                  ⁢                                                            r                  1                                ⁡                                  (                  k                  )                                            2                                                                                                                G            ~                    0          2                =                              (                                          -                                                      σ                    ~                                    0                  2                                            +                                                1                                                                                χ                      -                                                                                          ⁢                                                      ∑                                          k                      ∈                                              χ                        -                                                                                                                                            ⁢                                                                          ⁢                                                                                    r                        0                                            ⁡                                              (                        k                        )                                                              2                                                                        )                    ·                      2            /                          A              2                                                          (        8        )                                                      G            ~                    1          2                =                              (                                          -                                                      σ                    ~                                    1                  2                                            +                                                1                                                                                χ                      +                                                                                          ⁢                                                      ∑                                          k                      ∈                                              χ                        +                                                                                                                                            ⁢                                                                          ⁢                                                                                    r                        1                                            ⁡                                              (                        k                        )                                                              2                                                                        )                    ·                      2            /                          A              2                                                          (        9        )            wherein χ+ and χ− contain indices related to the first LTS symbols only, |χ| is the cardinality of the set χ and {tilde over (φ)} is the estimation of the parameter φ.
Once the noise variances and the channel gains are estimated, the SNRs defined in (2) or (3) are easily obtained. Furthermore, a reference value is introduced for each channel:Ti*=√{square root over ((A{tilde over (G)}i/√{square root over (2)})2/4+{tilde over (φ)}i2)}
This reference value is employed in the decision process.
Conventional FSK Detector
In order to give a comparison performance reference, a conventional FSK detector is detailed (see Algorithms for communications system and their applications,” N. Benvenuto and G. Cherubini, New York: Wiley, 2003).
The computation of the two decision values isμ+1(ρ0,ρ1)=ρ1 and μ−1(ρ0,ρ1)=ρ0  (10)while the decision rule is expressed by (6). Hereafter, this receiver is denoted with the label FSK.S-FSK Receivers Proposed in the Literature
Let I0(•) be the modified Bessel function of the first kind of order 0. As given in Schaub, when the transmitted symbol dk=−1, the amplitude probability density function hi|−(ρi) of the signal ri,k with iε{0,1} is:
                                          h                          0              ❘              -                                ⁡                      (                          ρ              0                        )                          =                                            2              ⁢                              ρ                0                                                    σ              0              2                                ⁢                                    I              0                        ⁡                          (                                                                    ρ                    0                                    ⁢                                      AG                    0                                    ⁢                                      2                                                                    σ                  0                  2                                            )                                ⁢                      exp            ⁡                          (                              -                                                                            ρ                      0                      2                                        +                                                                                            (                                                      AG                            0                                                    )                                                2                                            /                      2                                                                            σ                    0                    2                                                              )                                                          (        11        )                                                      h                          1              ❘              -                                ⁡                      (                          ρ              1                        )                          =                                            2              ⁢                              ρ                1                                                    σ              1              2                                ⁢                      exp            ⁡                          (                              -                                                      ρ                    1                    2                                                        σ                    1                    2                                                              )                                                          (        12        )            being h0|−(•) and h1|−(•) the amplitude probability density function of the envelop detector output on channels 0 and 1, respectively, when the symbol −1 has been transmitted.
On the other hand, when dk=+1, the amplitude probability density function hi|+(ρi) of the signal ri,k with iε{0,1} is:
                                          h                          0              ❘              +                                ⁡                      (                          ρ              0                        )                          =                                            2              ⁢                              ρ                0                                                    σ              0              2                                ⁢                      exp            ⁡                          (                              -                                                      ρ                    0                    2                                                        σ                    0                    2                                                              )                                                          (        13        )                                                      h                          1              ❘              +                                ⁡                      (                          ρ              1                        )                          =                                            2              ⁢                              ρ                1                                                    σ              1              2                                ⁢                                    I              0                        ⁡                          (                                                                    ρ                    1                                    ⁢                                      AG                    1                                    ⁢                                      2                                                                    σ                  1                  2                                            )                                ⁢                      exp            ⁡                          (                              -                                                                            ρ                      1                      2                                        +                                                                                            (                                                      AG                            1                                                    )                                                2                                            /                      2                                                                            σ                    1                    2                                                              )                                                          (        14        )            being h0|+(•) and h1|+(•) the amplitude probability density function of the envelop detector output on channels 0 and 1, respectively, when the symbol +1 has been transmitted. Note that hi,±(ρi)=0 when ρi<0. Assuming the symbols {−1,+1} to be transmitted with the same probability, the maximum likelihood decision may turn out to be the optimum decision rule. In particular, the decision rule (6) uses the following decision values:μ+1(ρ0,ρ1)=h0|+(ρ0)·h1|+(ρ1)μ−1(ρ0,ρ1)=h0|−(ρ0)·h1|−(ρ1)  (15)
Hereafter, this ideal receiver is denoted with the label Ideal. To practically implement the ideal receiver, the estimated channel and noise parameters may be used in the formulae from (11) to (15).
However, formulae from (11) to (15) are relatively complex and do not allow a direct practical implementation of the ideal decision rule. In order to obviate to this limitation, in Schaub, two suboptimal implementations are proposed.
First Prior Suboptimal Implementation
Let ri,k be quantized into the N intervals: Ii,1, Ii,2, . . . , Ii,N and let ρi fall into the intervals Ii,Ji, with iε{0,1}. The decision is taken according to (6) on the following decision values:μ+1(ρ0,ρ1)=v+1(ρ0)+v+1(ρ1)μ−1(ρ0,ρ1)=v−1(ρ0)+v−1(ρ1)  (16)where
                                          v                          ±              1                                ⁡                      (                          ρ              i                        )                          =                  log          ⁢                                    ∫                              I                                  i                  ,                                      J                    i                                                                                                                    ⁢                                                            h                                      i                    ❘                    ±                                                  ⁢                                                                  (                ρ                )                            ⁢                                                ⅆ                  ρ                                .                                                                        (        17        )            
The values vα(ρi), with iε{0,1} and αε{−1,+1}, are computed once the channel and the noise parameters are known. The practical implementation of (17) is, however, onerous. With a loss in performance, these decision values may be stored into a look-up table for a discrete set of channel parameters. Hereafter, this receiver is denoted with the label Real.1.
However, if the unbalancing factor x and the average signal-to-noise ratio Γ vary over a large range of values, a quite substantial amount of memory is used to help guarantee a negligible loss in performance compared to (17).
Second Prior Suboptimal Implementation
Due to the memory drawback of the Real.1 implementation, in Schaub a second suboptimal receiver is detailed. The detection rule is still given in (6), while the decision values are computed as follows:
A) if {tilde over (x)}>+τ*μ+1(ρ0,ρ1)=ρ1 and μ−1(ρ0,ρ1)=T1*  (18)B) if {tilde over (x)}<−τ*μ+1(ρ0,ρ1)=T0* and μ−1(ρ0,ρ1)=ρ0  (19)C) otherwise
                                                        μ                              +                1                                      ⁡                          (                                                ρ                  0                                ,                                  ρ                  1                                            )                                =                                    ρ              1                                                      G                ~                            1                                      ⁢                                  ⁢        and        ⁢                                  ⁢                                            μ                              -                1                                      ⁡                          (                                                ρ                  0                                ,                                  ρ                  1                                            )                                =                                    ρ              0                                                      G                ~                            0                                                          (        20        )            where τ*=4.77 dB. Hereafter, this real receiver is denoted with the label Real.2. In the cases A) and B) a conventional amplitude shift keying (ASK) receiver is realized (see for instance Benvenuto et al. and K. S. Shanmugan, “Digital and Analog Communication Systems,” New York: Wiley, 1979).
Unfortunately, the performance of the second sub-optimal approach is not satisfactory, in particular for values of the unbalancing factor x in the range [−10,10] dB.